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Hello and welcome back.

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Today we're talking about the R-squared.

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A very important concept
when it comes to evaluating the goodness

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of fit of our models.

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Now, in order to understand

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r squared we're going to need
to look at two versions of our chart.

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So here is our data set.

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And what we're going to do here is plot
the regression as we were doing before.

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And here's our data set again.

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And this time
we're going to just plot an average line.

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And we'll see that in a second.
So we'll start with the regression.

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Let's draw our regression line as usual.

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Let's project
vertically our data points onto it.

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And for each data point we're going to
look at the difference between the y

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the actual value
and y hat the predicted value.

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Now as we discussed before,
the way we built this line is

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we're minimizing this sum over here.

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And that is the ordinary
least squares method.

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Well actually this sum has a name.

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It's called the residual sum of squares.

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And it's indicated by this abbreviation.

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Now on the right here
we're going to draw an average line.

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And this is simply taking all of the
y values.

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So the actual values of our data
set the y values

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and taking their average.

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Again we're going to project
vertically our data points onto this line.

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And for each data point
we're going to look at y I.

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And here
we're going to calculate another total.

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And this one is called

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the total sum of squares
and is indicated by this abbreviation.

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And it's similar to the residual
sum of squares.

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But instead of using y
I hat we're looking at the difference

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between y I, the actual value
and y average the average value

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of of our data set.

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And now we can calculate r squared.

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R squared is defined
as one minus the ratio

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between the residual sum of squares
and the total sum of squares.

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Now let's pause here for a second
and discuss this a bit.

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So we know that we're
minimizing the residual sum of squares.

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We want to make it as small as possible.

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And from these two images
you can already see just by judging

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based on the lengths of these

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blue dotted blue dashed lines,
we can see that

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they're generally longer
on the right with averages

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and they're shorter
where the regression is.

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And that's because, we've designed
our line to minimize these lengths.

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So the sum of squared is smaller.

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And so what that means
is that the, residual

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sum of squares is usually in most cases
it's less than the total sum of squares.

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So the way to think about it is
in the total sum of squares on the right.

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You're just putting an average line.
You're not modeling anything.

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This is the most rudimentary thing
that we can do. is

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just put our average line and approximate
our data with that average line.

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Of course it will be.

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And it should be worse than any,

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thought out model that we create,
which is the example on the left.

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So thereby, unless our regression model
is facing the absolutely

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the wrong way, for example,
a downward slope on the left over here,

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if our model was sloping downwards,
then the residual

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sum of squares would be huge
because our model's just incorrect.

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But in all other cases, the residual
sum of squares is less than the total

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sum of squares.

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And what that means
is that the ratio is less than one.

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So r squared is somewhere between 0 and 1.

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And the better
our model fits the data, the smaller

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the residual sum of squares will be, and
that means the greater r squared will be.

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So here's a quick rule of thumb
for r squared.

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Now bear in mind
that it highly depends on the context.

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And this rule of thumb
is just for the practical tutorials

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that we're looking at
in this section of the course.

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And here we go.

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So if you have A1R squared of one
that's a perfect fit.

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That means, the residual
sum of squares is zero.

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And basically your line is going through

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all the data points,
which is virtually impossible.

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So it's very suspicious
if your r squared is about 0.9.

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That's a very good model.

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If your r squared is less than zero point.

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So it's not a great model.

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It's not the end of the world
but not great.

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If it's less than 0.4,
it's quite a terrible model.

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And if it's less than zero then
the model makes no sense for this data.

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As we discussed.

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So there we go.

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That's how the r squared works
a very important concept to understand

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because it's used
quite a lot to evaluate models.

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I look forward to seeing you next time.

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And until then, enjoy machine learning.